New upper and lower bounds for the upper incomplete gamma function

Abstract

Some new upper and lower bounds for the upper incomplete gamma function $\Gamma (a,x)$ are presented. Some of the bounds are given for all real $x>0$ and some are for only certain combinations of $a$ and $x$. A number of different methods are used to obtain these new bounds. In particular, rational function and perturbations of rational function bounds are presented. Some numerical comparisons are made with previously proposed bounds.